Control of Spontaneous Emission via a Single Elliptically Polarized Light in a Five-Level Atomic System

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1 Commun. Theor. Phys. 59 (213) Vol. 59, No. 5, May 15, 213 Control of Spontaneous Emission via a Single Elliptically Polarized Light in a Five-Level Atomic System ZHANG Duo ( ), 1,2, LI Jia-Hua (Ó Ù), 2, DING Chun-Ling (òë ), 3 and YANG Xiao-Xue ( ) 2 1 School of Electrical and Electronic Engineering, Wuhan Polytechnic University, Wuhan 4323, China 2 School of Physics, Huazhong University of Science and Technology, Wuhan 4374, China 3 School of Physics and Electronics, Henan University, Kaifeng 4754, China (Received December 17, 212; revised manuscript received April 1, 213) Abstract We investigate the features of the spontaneous emission spectra in a cold five-level atomic system coupled by a single elliptically polarized control field. We use wave function approach to derive the explicit and analytical expressions of atomic spontaneous emission spectra. It is shown that some interesting phenomena such as spectralline enhancement, spectral-line suppression, spectral-line narrowing, spectral-line splitting and dark fluorescence can be observed in the spectra by appropriately modulating the phase difference between the right-hand circularly (LHC) and left-hand circularly (RHC) polarized components of the elliptically polarized control field and the intensity of external magnetic field. The number of emission peaks, the positions of fluorescence-quenching points can be also controlled. Furthermore, we propose an ultracold 87 Rb atomic system for experimental observation. These investigations may find applications in high-precision spectroscopy. PACS numbers: 42.5.Gy, 32.8.Qk, d Key words: spontaneous emission, elliptically polarized field, spectral-line enhancement and narrowing 1 Introduction Spontaneous emission is a fundamental process resulting from the interaction of the atomic system with the environmental mode. [1 2] The usual method to modify and control the spontaneous emission of a multilevel atomic system is achieved by putting the atoms in frequency-dependent reservoirs, [3 4] in photonic crystals (PCs) [5 1] or in optical cavities. [11] For atoms in free space, atomic coherence and quantum interference are the basic mechanisms for efficiently controlling the spontaneous emission. [12 14] The potential applications for such a spontaneous-emission control in atomic systems via quantum interference and atomic coherence cover from lasing without inversion, [15 2] high-precision spectroscopy and magnetometry, [21 24] transparent high index materials, [25 28] spatial localization of atoms, [29 3] quantum information and computing, [31 33] and so on. The other alternative way to control atomic spontaneous emission is to coherently drive atoms with externally applied laser fields. In the past few decades, considerable effort has been devoted to investigate spontaneous emission in atoms driven by coherent fields. [34 4] Paspalakis and coworkers studied the quenching of spontaneous emission in an open V-type atom. [34] At the same time, phase-dependent effects in spontaneous emission spectra in a four-level atom were proposed and investigated in Refs. [38 39], where the relative phase of the two lasers was used to obtain partial cancellation, extreme linewidth narrowing, and total cancellation in the spontaneous emission spectrum. Recently, Wu et al. [41] studied the spontaneous-emission properties of a coherently driven four-level atom and showed a few interesting phenomena such as fluorescence quenching, spectral-line narrowing, spectral-line enhancement, and spectral-line elimination. In the following research, Li et al. [42] discussed the features of the spontaneous emission spectra of a five-level atomic system driven by two coherent laser fields and a radiofrequency/microwave field, and analyzed the reason in dressed-state picture. More recently, Li and coworkers [43] studied spontaneous emission of a four-level Y-type atomic system where the upper two levels are coupled by a microwave field. Qi [44] investigated the spontaneous emission from an inverted Y-type atomic system coupled by three coherent fields, and explained the spectral characteristics by the analytical expression with the corresponding physical parameters. To the best of our knowledge, no further theoretical or experimental work has been undertaken to study spontaneous emission properties and its control in such a five-level atomic system driven by a single elliptically polarized field and and a static magnetic field, which motivate the current work. At the same time, the spon- Supported by the National Natural Science Foundation of China under Grant Nos , and , the Doctoral Foundation of the Ministry of Education of China under Grant No , and by the National Basic Research Program of China under Grant No. 212CB zhangduo1@gmail.com huajia li@163.com c 213 Chinese Physical Society and IOP Publishing Ltd

2 No. 5 Communications in Theoretical Physics 595 taneous emission properties of an five-level atom in photonic band-gap materials have been discussed in detail in another paper. [45] In this paper, we investigate the spontaneous emission properties of a five-level double tripod-type atom controlled by an elliptically polarized control light and an external magnetic field. The V-type transitions from the two upper levels to one lower level are coupled by the left-hand circularly (RHC) polarized component of the control field, and the V-type transitions from the two upper levels to another lower level are coupled by the right-hand circularly (LHC) polarized component. The external magnetic field is used to create Zeeman shift between two lower levels. Using Laplace transform and final value theorem, [46] we derive the analytic expression of spontaneous emission spectra, and discuss the influence of polarization-dependent phase difference between the LHC and RHC polarized components of the elliptically polarized control field, the intensity of external magnetic field, and Rabi frequency of the elliptically polarized control field on atomic spontaneous emission properties. Meanwhile, the effect of atomic initial states on spontaneous emission spectra is also discussed. Compared with the previous schemes [34 39,41 44] for controlling the spontaneous emission behavior of a multilevel atomic system by using the external laser fields, the quantum interference between multiple decay channels in our considered system is caused by the two components of the elliptically polarized light beam in the presence of an external magnetic field. On the one hand, the phase difference between the two circularly polarized components can be used to change the degree of quantum interference, thus leading to some interesting spectral profiles. On the other hand, the applied magnetic field is more readily available and easier to control compared with an ordinary laser field, and this is another situation considered in this paper. In other words, these two aspects are the main advantages in the current work, which differ dramatically from the previous studies [34 39,41 44] on the manipulation of spontaneous emission spectra. These investigations provide more tunable parameters for us controlling the spontaneous emission properties. In particular, the ability to manipulate spontaneous emission of an atom by using an elliptically polarized light beam and a static magnetic field could find applications in high-precision spectroscopy. This paper is organized as follows: In Sec. 2, the theoretical model and corresponding equations are presented. We deduce the analytical expression for describing the spontaneous emission spectra of the system by use of the time-dependent Schrödinger equation. In Sec. 3, the spontaneous emission properties under different conditions are discussed. In Sec. 4, we provide a possible experimental realization of our scheme with cold 87 Rb atoms. Finally, we conclude with a brief summary in Sec Theoretical Model and Basic Formula We consider a five-level atomic system as shown in Fig. 1, with two excited states 3, 4 and three ground states 1, 2 and g. An elliptically polarized control field with a carrier frequency ω c and a wave vector k c can be regarded as a combination of the left- and rightcircularly polarized components, the left-circularly polarized component can be used to create electric dipole transitions from the excited states 4 and 3 to the ground state 1, and the right-circularly polarized component can couple the excited states 4 and 3 to 2 simultaneously. The elliptically polarized control field can be obtained by using a quarter-wave plate (QWP). [47] An initial vertically polarized control beam with intensity I and electric field amplitude E = 2I /(ε c) (ε is the permittivity of free space and c is the speed of light) becomes elliptically polarized after passing through the QWP that has been rotated by an angle θ (the polarization-dependent parameter), so the polarized control beam can be decomposed into E c = E + σ + +E σ, where E + = (E / 2)(cosθ+sin θ)e iθ and E = (E / 2)(cosθ sin θ)e iθ. Here, σ + and σ are the unit vectors of the RHC and LHC polarized basis, respectively. When θ = and π/2, we have E + = E, that is, the control beam is linearly polarized. When θ = π/4 (3π/4), we have E = (E + = ), that is, the control beam is right(left)-circularly polarized. QWP can change the strengths and phase difference of the two electric field components. Then, one half of Rabi frequencies become Ω c+ = µ 32 E + /(2 ) = µ 42 E + /(2 ) = Ω c (cosθ + sin θ)e iθ and Ω c = µ 31 E /(2 ) = µ 41 E /(2 ) = Ω c (cosθ sin θ)e iθ, here we assume µ 31 = µ 41 = µ 32 = µ 42 = µ (µ denotes the electric dipole moment between the corresponding transitions) and Ω c = µe /(2 2 ). While the transitions from the excited states 4 and 3 to the ground state g are assumed to be coupled by vacuum modes in the free space. The interaction Hamiltonian for the system composed of five-level atom, control fields and vacuum modes can be written as [48] (taking = 1) Ĥ = k + k g 3k e i(ω3g ω k)t b k 3 g g 4k e i(ω4g ω k)t b k 4 g + Ω c e i( c+ B δ)t Ω c e i( c+ B+δ)t Ω c+ e i( c B δ)t Ω c+ e i( c B+δ)t H.c., (1) where the rotating-wave approximation (RWA) and electric-dipole approximation (EDA) have been made. H.c. means Hermitian conjugation and g jk (j = 3, 4) is the coupling constant between the atomic transition j g with the mode k of the radiation field. Without loss of generality, both coupling constants are assumed to be real. ω nm = ω n ω m is the energy separation of the states n and m, the notation c = (ω 31 +ω 41 )/2 ω c B is the detuning of the transitions from the corresponding laser frequencies [see Fig. 1]. While ω 41 ω c = c + B + δ, ω 31 ω c = c + B δ, ω 42 ω c = c B + δ and ω 32 ω c = c B δ represent the detunings of the

3 596 Communications in Theoretical Physics Vol. 59 external control field from the corresponding atomic transition, where δ = ω 43 /2 = (ω 41 ω 31 )/2 = (ω 42 ω 32 )/2 is a half of frequency difference of the upper two levels. k = (ω 3g + ω 4g )/2 ω k is the detuning of the spontaneously emitted photon with frequency ω k from the middle of the two upper levels. B is the Zeeman shift of levels 1 and 2 in the presence of the magnetic field. b k (b k ) is the annihilation (creation) operator for the k-th reservoir mode with frequency ω k, k here represents both the momentum vector and the polarization of the vacuum mode. Fig. 1 Schematic diagram of a driven five-level atomic system, which consists of three lower levels 1, 2, g, and two excited levels 3 and 4. A left(right)-hand circularly polarized component Ω c (Ω c+) of the elliptically polarized control field induce the transitions 3 and 4 to the lower level 1 ( 2 ). The symbol c is the frequency detunings of the control field, see text for details. By applying a uniform magnetic field parallel to the propagation direction of the elliptically polarized control beam, the degeneracy among the ground states 1 and 2 is lifted, where an amount B stands for the Zeeman shift. k = (ω 3g + ω 4g)/2 ω k is the detuning of the spontaneously emitted photon with frequency ω k from the average atomic transition frequency (ω 31 + ω 41)/2. In the interaction picture, the state vector of the atomic system at time t can be expressed as Ψ(t) = [a 1 (t) 1 + a 2 (t)e i2 Bt 2 + a 3 (t)e i( c+ B δ)t 3 + a 4 (t)e i( c+ B+δ)t 4 ] {} + k a k (t) g 1 k, (2) where {} represents the vacuum of electromagnetic field, and 1 k denotes that there is one photon in the k-th vacuum mode. Substituting Eq. (2) into the Schrödinger equation i Ψ(t) / = Ĥ Ψ(t), we get the coupled equations of motion for the probability amplitudes as a 1 (t) = iω c a 3 (t) iω c a 4 (t), (3) a 2 (t) = i2 B a 2 (t) iω c+ a 3(t) iω c+ a 4(t), (4) a 3 (t) [ = i ( c + B δ) i Γ ] 3 a 3 (t) 2 a 4 (t) a k (t) iω c a 1 (t) iω c+ a 2 (t), (5) [ = i ( c + B + δ) i Γ ] 4 a 4 (t) 2 iω c a 1 (t) iω c+ a 2 (t), (6) = ig 3k e i( c+ B k)t a 3 (t) ig 4k e i( c+ B k)t a 4 (t), (7) where Γ j = 2π g jk 2 D(ω k ) (j = 3, 4) is the spontaneousdecay rate from the excited level j to the ground level g, and D(ω k ) is the vacuum-mode density at frequency ω k in the free space. Carrying out the Laplace transformations [46] ã j (s) = e st a j (t)dt for Eqs. (3) (6) and integrating Eq. (7) with respect to t, where s is the time Laplace transformation variable, we can obtain the following results where sã 1 (s) a 1 () = iω c ã3(s) iω c ã4(s), (8) sã 2 (s) a 2 () = i2 B ã 2 (s) iω c+ã 3 (s) iω c+ã4(s), (9) [ sã 3 (s) a 3 () = i ( c + B δ) i Γ ] 3 ã 3 (s) 2 iω c ã 1 (s) iω c+ ã 2 (s), (1) [ sã 4 (s) a 4 () = i ( c + B + δ) i Γ ] 4 ã 4 (s) 2 iω c ã 1 (s) iω c+ ã 2 (s), (11) a k = ig 3k ig 4k t t e iw3t a 3 (t )dt e iw3t a 4 (t )dt, (12) w 3 = c + B k. a j () (j = 1 4) is the probability amplitude at the initial time t =. Equations (8) (11) can be solved directly in terms of a 1 (), a 2 (), a 3 (), and a 4 (). Therefore, the solution to the probability amplitude ã 3 (s) and ã 4 (s) can be found as ã 3 (s) = (iω c /s)a 1 () + [iω c+ /(s + i2 B )]a 2 () + (1 + B/A 4 )a 3 () (B/A 4 )a 4 () A 3 + B + (A 3 /A 4 )B, (13)

4 No. 5 Communications in Theoretical Physics 597 ã 4 (s) = (iω c A 3 /sa 4 )a 1 () [(iω c+ A 3 )/(s + i2 B )A 4 ]a 2 () (B/A 4 )a 3 () [(A 3 + B)/A 4 ]a 4 () A 3 + B + (A 3 /A 4 )B, (14) where A 3 = s + i( c + B δ) + Γ 3 /2, A 4 = s + i( c + B + δ) + Γ 4 /2, and B = Ω c 2 /s + Ω c+ 2 /(s + i2 B ), respectively. According to the above Eq. (12), we obtain a k in the long-time limit as a k (t ) = ig 3k e iw3t a 3 (t )dt ig 4k e iw3t a 4 (t )dt = ig 3kã3(s = iw 3 ) ig 4kã4(s = iw 3 ), (15) where ã j (s) (j = 3, 4) is the Laplace transformation of a j (t). The spontaneous emission spectra S( k ) are proportional to a k (t ) 2. We neglect the interference terms between two sets of dressed states corresponding to the two bare states 3 and 4, and write the spontaneous emission spectra as S( k ) = Γ 3 2π ã 3(s = iw 3 ) 2 + Γ 4 2π ã 4(s = iw 3 ) 2. (16) Under the conditions a 1 () = a 2 () =, a 3 () = a 4 () = 1/ 2 and Γ 3 = Γ 4 = γ, the spontaneous emission spectra S( k ) in Eq. (16) can be explicitly expressed in the following form S( k ) = γ {1 + [i( k δ) + γ/2]/[i( k + δ) + γ/2] 2 }/2 2π i( k δ) + γ/2 + M + N + {[i( k δ) + γ/2]/[i( k + δ) + γ/2]}(m + N) 2, (17) where M = Ω c 2 (1 sin 2θ) i( k B c ), N = Ω c 2 (1 + sin 2θ) i( k + B c ). Equation (17) is the most important result of this paper. It is not difficult to find from Eq. (17) that the analytical expressions for the spontaneous emission spectra S( k ) is sensitively dependent on the controllable parameters of the system such as the polarization-dependent parameter θ, the Zeeman shift B or the magnetic field intensity B ( B B) and Rabi frequency of the control field. As a consequence, we can control the spontaneous emission of the five-level double tripod-type atomic system by adjusting these system parameter under proper conditions. 3 Numerical Results and Analysis The steady state behavior of spontaneous emission of our system can be evaluated using partially dressed states method. [49] The upper level 4 ( 3 ) is connected with the lower levels 1 and 2, and interacts with level 3 ( 4 ) by two components of elliptically polarized field Ω c and Ω c+, which makes bare state 4 ( 3 ) split into four dressed states. Since each transition corresponding to one peak in the spectra, the spontaneous emission spectra are expected to have eight peaks corresponding to the two sets of dressed states decay to the final state. However, the spectra line can be eliminated or fuse into each other under certain conditions. For the system we investigated, only four peak can be observed because the peaks corresponding to bare state 3 should overlap with the peaks corresponding to bare state 4 due to the same denominator of Eqs. (13) and (14) (peak arises when denominator tends to zero). Now, we present numerical results of the spontaneous emission spectra based on Eqs. (16) and (17). All parameters used in the following calculations are scaled by γ, which should be in the order of MHz for rubidium atoms. First of all, we will analyze how the polarizationdependent phase difference between the two circularly polarized components of the control beam modifies spontaneous emission spectra via the numerical calculations. Figure 2 shows the spontaneous emission spectra S( k ) versus the detuning k when the atom is initially prepared in an equal superposition of the two levels 3 and 4 for different phase difference. It is shown that the phase difference of two circularly polarized components of the control field determines both width of the spectral line and the number of emission peaks. The insets display 1 typical forms of polarized control beam. For the case that θ = [see Fig. 2(a)], we have the result E + = E, that is, the control beam is linearly polarized, and the control beam s intensity distribution among its σ + and σ components are the same. Thus the spontaneous emission spectra exhibit an symmetric four-peak structure with two fluorescence-quenching points. It can be easily seen from Eq. (17) that when k = B + c or c B, the denominator of S( k ) tends to infinity, while the numerator of S( k ) is limited, as a result two fluorescence-quenching points arise. When the phase difference θ increase to π/6 [see the black dashed line in Fig. 2(b)], the control beam is right-handed elliptically polarized, we can find that the right side peak and the left central peak are suppressed slightly, while the right central peak is enhanced obviously and moves toward the right side peak. With a gradual increase of phase difference θ to 2π/9 [see the red solid line in Fig. 2(b)], the right peak near the center evolves into an ultranarrow line, which should be attributed to quantum interference of different

5 598 Communications in Theoretical Physics Vol. 59 decay channels coming from dressed states. However, in the case of θ = π/4 [see Fig. 2(c)], we have E = ; that is to say, the control beam is right-handed circularly polarized, which makes level 3 ( 4 ) split to three dressed states. As a result, the right central peak is completely suppressed and spontaneous emission spectra show a triple-peak structure. With phase difference θ further increasing to 5π/18 (π/3 or π/2), the spontaneous emission spectra show the same shape with θ = 2π/9 (π/6 or ) [see Figs. 2(b) and 2(a)]. When θ further increases to θ = 5π/9, the control beam is left-handed elliptically polarized, the emission spectra show a slightly enhanced left central peak and a slightly suppressed right central peak. More interestingly, as shown in Figs. 2(e) and 2(f), we can observe that the new spectra are just the mirror inversion of Fig. 2(b) and Fig. 2(c) when phase difference varied from 2π/3 to 3π/4. Similarly, for the case of phase difference θ = 7π/9 (5π/6 or 17π/18), the spontaneous emission spectra show the same shape as 13π/18 (2π/3 or 5π/9) [see Figs. 2(e) and 2(d)]. Fig. 2 (Color online) The spontaneous emission spectra S( k ) (in units of γ 1 ) for Ω c = 1.5γ, B = 2γ, c =, δ = 2γ, a 1() = a 2() = and a 3() = a 4() = 1/ 2. The insets show ten typical forms of polarized control field. (a) θ =, π/2; (b) θ = π/6, π/3 (black dashed line), θ = 2π/9, 5π/18 (red solid line); (c) θ = π/4; (d) θ = 5π/9, 17π/18; (e) θ = 2π/3, 5π/6 (black dashed line), θ = 13π/18, 7π/9 (red solid line); (f) θ = 3π/4. The agreement between the analytical solution and the numerical calculation is good. From the above expression (17), we can find that the spontaneous emission spectra S( k ) is strongly dependent on the laser-polarizationdependent phase θ and also is a periodical function of the relative phase θ with a period π. Furthermore, it can be easily seen from the expression (17) that the relations [S( k )] θ = [S( k )] π/2 θ = [S( k )] 3π/2 θ holds, which can well explain the numerical results of Fig. 2. The interesting phenomena in the spontaneous emission spectra discussed above can be qualitatively attributed to quantum interference of different decay channels. We can see that when populations are initially prepared in superposition of the two levels 3 and 4, they decay to the ground state g through the following processes: 4 ( 3 ) g, 4 ( 3 ) 1 4 ( 3 ) g, 4 ( 3 ) 1 3 ( 4 ) g, 4 ( 3 ) 2 4 ( 3 ) g, 4 ( 3 ) 2 3 ( 4 ) g. Due to destructive quantum interference between competitive pathways, the spontaneous emission will have fluorescence quenching points and spectral-line suppression. While, constructive quantum interference between different pathways results in spectral-line enhancement and narrowing. In the following, we consider the effect of external magnetic field on the spontaneous emission spectra for the atom is initially prepared in a superposition of the two levels 3 and 4. In Fig. 3, we plot the spectra S( k ) versus the detuning k for different intensities of external mag-

6 No. 5 Communications in Theoretical Physics 599 netic field. As shown in the figure, the shape of spectral line and location of dark line depend sensitively upon the strength of the applied magnetic field. In the absence of external magnetic field (i.e., B = as shown in Fig. 3(a), which corresponds to B = ), the emission spectra display a symmetric double peaks with normal width and one fluorescence-quenching point located at k =, which can be explained as follows. When B =, the detuning of atomic transition 1 and 3 ( 4 ) from laser frequency ω c is the same with the other detuning of transition 2 between 3 ( 4 ) (i.e., c + B δ = c B δ ( c + B +δ = c B +δ)), the two-photon resonance condition is fulfilled, the so-called dark state or coherent population trapping (CPT) state can be formed, so emission spectra exhibit two peaks. While, when external magnetic field is presented and Zeeman shift B, the CPT condition is deviated, we can observe two extra spectral lines as shown in Figs. 3(b) 3(d). From Fig. 3(b), we can found that a small peak and a narrow spectral line appear at the center of emission spectra in the case of B = γ. Moreover, the height of the right side peak becomes lower. When the external magnetic field B gradually increases to the corresponding Zeeman shift B = 2γ, the central peak and central ultranarrow line are enhanced obviously. Meanwhile, the location of two dark line moves away from each other (dark line exists at k = c ± B ) [see Fig. 3(c)]. More interestingly, with further increase external magnetic field B corresponds to B = 4γ, the ultranarrow line near the center evolves into a normal width peak. While two side peaks evolve into two ultranarrow lines simultaneously as shown in Fig. 3(d). Fig. 3 The spontaneous emission spectra S( k ) (in units of γ 1 ) for Ω c = 1.5γ, c =, δ = 2γ, θ = 2π/9, a 1() = a 2() = and a 3() = a 4() = 1/ 2. (a) B = ; (b) B = γ; (c) B = 2γ; (d) B = 4γ. In order to further explore the effect of the intensity of control field on spontaneous emission spectra, we also plot S( k ) versus the detuning k in Fig. 4. We can clearly see from Fig. 4(a) that only two symmetric peaks exist in emission spectra in the absence of the control field [see black dashed line in Fig. 4(a)], which should be attributed to spontaneous decay from levels 3 and 4 to level g. Whereas in the presence of control field (Ω c =.5γ), the left peak splits to two peaks and the right peak splits to an ultranarrow line and a normal width peak [see red solid line in Fig. 3(a)]. Increasing the intensity of control field and considering the case of Ω c = γ and Ω c = 2γ [see Figs. 4(b) and 4(c)], the central peak and ultranarrow line are gradually suppressed. However, the two side peaks are slightly enhanced and move away from each other. Specifically, when the external control field intensity continues to increase (e.g., Ω c = 4γ in Fig. 4(d)), the central small peak and ultranarrow line are nearly suppressed completely and side peaks continuously move away from the center, which should be attributed to quantum interference of different decay channels.

7 6 Communications in Theoretical Physics Vol. 59 Fig. 4 (Color online) The spontaneous emission spectra S( k ) (in units of γ 1 ) for B = 2γ, c =, δ = 2γ, θ = 2π/9, a 1() = a 2() = and a 3() = a 4() = 1/ 2. (a) Ω c = (black dashed line), Ω c =.5γ (red solid line); (b) Ω c = γ; (c) Ω c = 1.5γ; (d) Ω c = 3γ. Fig. 5 The spontaneous emission spectra S(δ k ) (in units of γ 1 ) for four different atom initial states: (a) a 1() = 1, a 2() = a 3() = a 4() = ; (b) a 1() = a 3() = a 4() =, a 2() = 1; (c) a 1() = a 2() = a 4() =, a 3() = 1; (d) a 1() = a 2() = a 3() =, a 4() = 1; (e) a 1() = a 2() = 1/ 2, a 3() = a 4() = ; (f) a 1() = a 2() =, a 3() = a 4() = 1/ 2 in the case of Ω c = 1.5γ, B = 2γ, c =, δ = 2γ and θ = 2π/9.

8 No. 5 Communications in Theoretical Physics 61 It is well known that the shape of the spontaneous emission spectra depends strongly on the atomic initial states. Finally, we investigate the influence of system initial states on spontaneous emission spectra S( k ) by employing the state preparation technique demonstrated previously in experiments (e.g., see Ref. [38] and references therein for details). For the case that a 1 () = 1 and a 2 () = a 3 () = a 4 () =, only an ultranarrow spectral line can be observed in the spontaneous emission spectra [see Fig. 5(a)]. For the case that a 2 () = 1 and a 1 () = a 3 () = a 4 () =, we can observe three normal width peaks and a small narrow line with one fluorescencequenching point located at k = c + B [see Fig. 5(b)]. For the case that a 3 () = 1 and a 1 () = a 2 () = a 4 () =, the emission spectra still exhibit three normal width peaks and a narrow line, while the right peak is higher than the left and the fluorescence-quenching point is located at k = c B [see Fig. 5(c)] compared with the case of a 2 () = 1 and a 1 () = a 3 () = a 4 () =. While, under the conditions of a 4 () = 1 and a 1 () = a 2 () = a 3 () =, two normal height peaks appear at the left side of k and a lower peak exists at the right [see Fig. 5(d)]. When the initial probability amplitudes subject to a 1 () = a 2 () = 1/ 2 and a 3 () = a 4 () =, three lower peaks and a greatly enhanced ultranarrow spectral line occur in the spontaneous emission spectra with two fluorescence-quenching points [see Fig. 5(e)]. With a 3 () = a 4 () = 1/ 2 and a 1 () = a 2 () =, we show the circumstance where no fluorescence-quenching points exist in the spontaneous emission spectra. The spectra also exhibit three normal width peaks and an ultranarrow line as shown in Fig. 5(f). 4 Possible Experimental Realization of the Proposed Scheme Before ending this section, let us briefly discuss the possible experimental realization of our proposed model of Fig. 1 by means of alkali-metal atoms, external magnetic field and elliptically polarized light. For example, we consider D 2 line of the cold 87 Rb atoms (nuclear spin I = 3/2) as a possible candidate. [51] The designated states and the decay rates can be chosen as follows: 1 = 5S 1/2, F = 1, m = +1, 2 = 5S 1/2, F = 2, m = 1, 3 = 5P 3/2, F = 1, m =, 4 = 5P 3/2, F = 2, m =, g = 5S 1/2, F = 1, m =, and Γ 3 = Γ 4 = 2π 6 MHz, respectively. In this case, the wavelength of elliptically polarized control field is about 78.2 nm, which can be obtained through the following ways. Before applying to 87 Rb atoms, the control beam passes through a half-wave plate (HWP) followed by a QWP. [47] The HWP makes the control beam vertically polarized, and the QWP has been rotated by an angle θ to control the polarization of the incoming vertically polarized control beam. After passing through the QWP, the control beam becomes elliptically polarized light which contains two components Ω c+ and Ω c. Alternatively, in order to eliminate the Doppler broadening effect, atoms should be trapped and cooled by the magneto-optical trap (MOT) technique. Finally, we briefly address the requirement of external magnetic field. For instance, when taking the Zeeman shift B = 2γ = 2 2π 6 MHz in our numerical calculations, according to the relationship B = mµ B g F B with the magnetic quantum number m = 1, the Landé factor g F = 1/2 and the Bohr magneton µ B = J/T, the magnetic field strength B T = 17.2 G is needed and can be easily fulfilled in the experiment. 5 Conclusion In summary, we have theoretically investigated the spontaneous emission spectra of a five-level atom controlled by an elliptically polarized field and an external magnetic field. A wave function approach is used to derive the explicit and analytical expressions of atomic spontaneous emission spectra. The results clearly show that, by choosing appropriate parameters of the coupled system, such as the intensity of external magnetic field, Rabi frequency of elliptically polarized control field and the initial probability amplitudes, we can observe a few interesting phenomena in the spontaneous emission spectra, such as spectral-line narrowing, spectral-line enhancement and spectral-line suppression, spectral-splitting and dark fluorescence. Specially, the spontaneous emission spectra are sensitively dependent on the polarization-dependent phase difference between the LHC and RHC polarized components of the elliptically polarized control field. So we can manipulate the spontaneous emission effectively by adjusting appropriately the elliptically polarized control field. The origin of the spectral characteristics can be attributed to quantum interference of different decay channels which is agreed with the analytical expressions of the spontaneous emission spectra. Moreover, since no rigorous condition is required in our scheme, the numerical results for atomic spontaneous emission can be observable in realistic experiments by using the 87 Rb atom in MOT where the atomic temperature can be decreased to several tens of µk so that the Doppler broadening effect can be effectively eliminated. Acknowledgments The authors would like to thank Prof. WU Ying for his encouragement and helpful discussion.

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